Is this the correct thinking process?
I am thinking about a proposition that says a set is closed if it contains all of its limit points: I am just looking at –> (this direction)
I suppose that the closed subset is called $X$ of metric space $M$ and that an arbitrary limit point $a$ is not in the set. Since $X$ is closed then the complement is open. Since the complement is open, there exists an open ball of any point in the complement. If we look at the limit point in the complement we see that it does not contain an infinite number of points in $X$ since the points it contains are in the complement of $X$. So, then $a$, the limit point, has to be in $X$. So a closed set contains all of its limit points?
Am I thinking on the right tract?
Thanks!
Best Answer
You are right. The proof can be simple as this:
Suppose that $x\notin X$ and $x$ is the limit point of $X$. Pick $U=X^{c}$ which is obvious contains $x$ and we have $U\cap X=\emptyset$. A contradiction!